A five-degree model, which reproduces faithfully the sequence of bifurcations and the type of solutions found through numerical simulations of the three-dimensional Boussinesq thermal convection
equations in rotating spherical shells with fixed azimuthal symmetry, is derived. A low Prandtl number fluid of s=0. 1 subject to radial gravity, filling a shell of radius ratio ¿=0.35, differentially heated, and with non-slip boundary conditions, is considered. Periodic, quasi-periodic, and temporal chaotic flows are obtained for a moderately small Ekman number, E=10-4,andatsupercritical Rayleigh numbers of order
Ra~O(2Rac). The solutions are classified by means of
frequency analysis and Poincaré sections. Resonant phase locking on the quasi-periodic branches,as well as a sequence of period doubling bifurcations, are also detected.

We study the hyperbolic system of equations of the so-called linear transport model in a true moving bed chromatography device with four ports. By using methods based on a suitable energy functional we show that all solutions approach exponentially a unique steady-state solution. Then, with the use of asymptotic analysis techniques we calculate the limit profiles of these steady-state solutions when the mass transfer coefficient between the liquid and solid phases tends to infinity. Along this singular limit sharp boundary layers appear near some ports. We are able to obtain explicit and simple formulas for these limit profiles.

We consider the dynamics of coorbital motion of two small moons about a large planet which have nearly circular orbits with almost equal radii. These moons avoid collision because they switch orbits during each close encounter. We approach the problem as a perturbation of decoupled Kepler problems as in Poincaré's periodic orbits of the first kind. The perturbation is large but only in a small region in the phase space. We discuss the relationship required among the small quantities (radial separation, mass, and minimum angular separation). Persistence of the orbits is discussed.

We consider the dynamics of coorbital motion of two small moons about a large planet which have
nearly circular orbits with almost equal radii. These moons avoid collision because they switch
orbits during each close encounter. We approach the problem as a perturbation of decoupled Kepler
problems as in Poincar ́
e’s periodic orbits of the first kind. The perturbation is large but only in a
small region in the phase space. We discuss the relationship required among the small quantities
(radial separation, mass, and minimum angular separation). Persistence of the orbits is discussed.